First slide

Theory of equations

Question

The equation formed by decreasing each root of ${ax}^{2} + bx + c = 0$ by 1 is $2 x^{2} + 8 x + 2 = 0,$ then

Moderate

A

$a = - b$
B

$b = - c$
C

$c = - a$
D

$b = a + c$

Solution

$α, β$ be the roots of ${ax}^{2} + bx + c = 0$
$∴ α + β = - b / a, αβ = c / a$
Now roots are $α - 1, β - 1$
Their sum, $α + β - 2 = (- b / a) - 2 = - 8 / 2 = - 4$
Their product, $(α - 1) (β - 1) = αβ - (α + β) + 1$
$\frac{c}{a} + \frac{b}{a} + 1$ =1
$∵$ New equation is $2 x^{2} + 8 x + 2 = 0$
$also c + b = 0 \Rightarrow b = - c$

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